Statistical physics and light-front quantization Jörg Raufeisen (Heidelberg U.) JR and S.J. Brodsky, Phys. Rev. D70, 085017 (2004) and hep-th/0409157
Introduction: Dirac s Forms of Hamiltonian Dynamics Front form: Define initial conditions on a light-like hypersurface. The evolution in light-cone time x + is then generated by the Poincare generator P, i Ψ = P Ψ Scalar product (x + = t + z = 2x ): x p = 1 2 (x+ p +x p + ) x p. Mass shell condition: P + P P 2 = M 2 In the Front Form, 7 out of 10 Poincare generators are independent of the interaction, among them are Ĵ3, K3, P + and P. The Light Front Hamiltonian P, of course, contains interaction terms.
The QCD Phase Diagram (according to MIT) So far, all of light-front quantization takes place at zero temperature and zero density. However, there is much more: RHIC (and LHC) probe strongly interacting matter under conditions similar to those in the early universe. T RHIC QGP (figure: M. Alford) = color superconducting crystal? The region with large baryon chemical potential and low T cannot be explored in the lab, but the core of neutron stars might be color superconducting. gas nuclear liq compact star In addition, systems such as the wavefunction of a large nucleus may also have statistical features. (Iancu et al. hep-ph/0410018) Advantage of the Front Form: K 3 independent of interaction frame independent distribution functions Fewer problems with fermions in numerical approaches (DLCQ) CFL µ
The Statistical Operator ŵ In statistical physics, the light-front Schrödinger equation, ı Ψ = P Ψ, is replaced by the Light-Front Liouville Theorem : [ i ŵ = P, ŵ]. For pure states ŵ = h h. In equilibrium, [ P, ŵ] = 0 ŵ is a function of those Poincaré generators that commute with P. The entropy operator ln ŵ is a linear function of these generators. Thus ŵ = exp { β(u µ P µ ωĵ3 i µ i Qi ) } u µ is the four velocity of the system and ω its angular velocity. µ i and Q i are chemical potentials and conserved charges, respectively. In thermodynamics, one usually works in the rest frame, so that u µ P µ ωĵ3 = P 0. There is no frame with u µ P µ ωĵ3 = P. Thus, the equal time energy is special among the ten Poincaré generators, even on the light front.
Statistical Ensembles The microcanonical ensemble, ŵ 0 MC δ(p + i p + i )δ(p i p i )δ(2) (P i p i, ) is appropriate for small systems, such as single hadrons. The (grand) canonical ensemble (let P = 0 ), ŵ = { [ ]} exp β u + M h 2 2P + + u p + i µq φ n/h (X)φ n 2 /h (X ) nx n X h n,n X,X i is appropriate for large systems, such as nuclei and neutron stars. Here: h=hamiltonian eigenstate, n=fock-state number, X=all other variables. Given a set of LC wavefunctions φ n/h (X) (obtained from DLCQ), one can calculate any expectation value. Momenta and charges are extensive quantities with a conjugate intensive quantity. This is not the case for Mh 2 and the LC momentum fractions x.
Thermodynamics All known thermodynamic relations hold in Light-Front Quantization, e.g. F = T ln T rŵ. Defined this way, the Free Energy F is a Lorentz scalar. So is the entropy, S = ( F T Equilibrium conditions: The five independent parameters T, u κ, µ are Lagrange multipliers that keep the mean value of a quantity constant, while entropy is maximized. The values of T, u κ, µ are independent of the system size. Two systems (1) and (2) are in equilibrium with each other, if T (1) = T (2), u (1) κ = u (2) κ, µ (1) = µ (2). No macroscopic motion is possible in equilibrium. (At least in the absence of vortices.) The entropy of an ideal gas is maximized for Bose-Einstein and Fermi-Dirac distributions, n(u µ k µ ) = [exp(βu κ k κ βµ) ± 1] 1 [exp(βu κ k κ + βµ) ± 1] 1. ) V.
Light-Front Quantization of the Fermi Field The 2-component fermion field operators in the Schrödinger picture are expanded as Ψ(r) = d 3 k (2π) λ 3 2 k + Θ(k+ ) { b(k, λ)χλ e ık r + d (k, λ)χ λ e +ık r}, Ψ (r) = d 3 k { b (2π) λ 3 2 k + Θ(k+ ) (k, λ)χ e+ık r λ + d(k, λ)χ e ık r} λ, with σ 3 χ λ = λχ λ, r = (r, r ), k = (k +, k ). The creation and annihilation operators obey the anticommutation relations { } { b(k, λ), b (k, λ )} = d(k, λ), d (k, λ ) = (2π) 3 2k + δ (3) (k k )δ λ,λ, so that the anticommutator of the dynamical spinor components at equal light-cone time r + reads α, β {1, 2} { Ψα (r), Ψ } β (r ) = δ α,β δ (3) (r r ). The entire theory can be formulated in terms of 2-component spinors, but is non-local along the light-cone.
In Medium Green s Functions of a Fermion The (time-ordered) Green s function is defined in terms of Heisenberg field operators as ıg α,β (r 1, r 2 ) = T + ψα (r 1 ) ψ β (r 2) = ψ α (r 1 ) ψ β (r 2) Θ(r + 1 r+ 2 ) ψ β (r 2) ψ α (r 1 ) Θ(r + 2 r+ 1 ). The average... has to be taken with the appropriate ensemble. In addition, retarded and advanced Green s functions are defined as the anticommutators { ıg R α,β(r 1, r 2 ) = ψα (r 1 ), ψ } β (r 2) Θ(r 1 + r+ 2 ) { ıg A α,β (r 1, r 2 ) = ψα (r 1 ), ψ } β (r 2) Θ(r 2 + r+ 1 ). The free Green s functions in momentum space read G (0)R,A α,β (k) = δ α,β k + G (0) α,β (k) = δ α,β k 2 m 2 ± ı0sgn(uk), (P k+ k 2 m 2 ısgn(uk)πtanh ( ) ) uk k + δ(k 2 m 2 ). 2T There is only one derivative in the numerator, leading to only one pair of fermion doublers on the lattice.
Separation of Quark and Antiquark Distributions Knowledge of the Green s function ıg α,β (r 1, r 2 ) = ψ α (r 1 ) ψ β (r 2) Θ(r 1 + r+ 2 ) ψ β (r 2) ψ α (r 1 ) Θ(r 2 + r+ 1 ) enables one to calculate all quantities pertaining to a single particle. In the limit r + 0 ±, G α,β (r 1, r 2 ) yields the one-particle density matrices for quarks and antiquarks, from which the expectation value of any single-particle operator can be calculated, F α,β (r + ) = d 3 r 1 d 3 (1) [ r 2 f β,α qα,β (r 1, r 2 ) + q α,β (r 1, r 2 ) ] δ (3) (r 1 r 2 ) d3 r [ q α,α (r, r) + q α,α (r, r) ], with F α,β (r + ) = d 3 r ψ α(r) f β,γ ψγ (r). The density matrix for quarks is given by (R = (r 1 + r 2 )/2, r = r 1 r 2 ) : q α,β (p +, R, r ) = 1 dr e +ıp+ r /2 4π ψ β (R + r 2 ) ψ α (R r 2 ) r + 1 =r+ 2 Similarly for antiquarks, q α,β (p +, R, r ) = ı 4π dr e +ıp+ r /2 G α,β (r + 1 0+, r 1, r + 2 = 0, r 2).
The Light-Front Density Matrix and GPDs q α,β (p +, R, r ) contains all information about unpolarized (q), longitudinally polarized ( L q) and transverse spin ( T q) distributions, q α,β (p +, R, r ) = q(p +, R, r )δ α,β + L q(p +, R, r )σ 3 α,β + T q (1) (p +, R, r )σ 1 α,β + T q (2) (p +, R, r )σ 2 α,β. There are actually 8 independent density matrices in QCD, 2 of them T-odd. The probability distribution for quarks is obtained by setting r = 0. q α,β (p +, R, r = 0 ) contains all information about the impact parameter dependence of PDFs, with b = R. These can be identified with GPDs at ζ = 0. (M. Burkardt) The quark density matrix contains terms that are off-diagonal in Fock-space (b d and bd.) These contribute to DVCS in different kinematical domains, γ (q) γ(q ) γ (q) γ(q ) k, λ k, λ k, λ k, λ P P ζ X 1 P P 0 X ζ
Covariant Introduction of a Chemical Potential In equal-time statistics at finite density, Heisenberg operators are defined using the Hamiltonian P 0 = P 0 µ Q. In light-front statistics, the effective Hamiltonian is Ĥ = u κ P κ µ Q. This operator propagates the system along its worldline. In equilibrium, u κ is a constant independent of time and position. The translation generators, that propagate the along trajectories with constant Q are then P κ = P κ µ Qu κ. The chemical potential modifies the translation generators like a gauge field. We therefore define Heisenberg operators as, ψ α (r) = e ı bp r + /2 Ψα (r)e ı bp r + /2, ψ α(r) = e ı b P r + /2 Ψ α (r)e ı b P r + /2.
Calculation of Thermodynamic Quantities Light-front QCD at finite temperature and density combines apparently different fields of physics. The Green s function is not only related to PDFs in the limit r + ±0, one can also calculate thermodynamic quantities from G: Obtain the net charge Q = q q as a function of µ, T and V from G. Integrate to obtain the grand-canonical potential. ( ) Ω µ V,T = Q Given a complete DLCQ solution of a theory, one knows the entire set of light-cone wavefunctions. This would enable one to also calculate all finite temperature properties of that theory. Advantage of DLCQ over Lattice: No problems with dynamical fermions or finite µ. That makes DLCQ the only known first principle approach applicable to high density field theory.
Summary Light-Front quantization has been generalized to finite temperature and density. Even on the Light-Front, the statistical operator is the exponential of the equal time energy in the rest frame of the system, { ŵ = exp β(u κ P κ ωĵ3 µ Q) }. The special meaning of the equal-time energy follows directly from the Poincaré algebra. At T = 0, the System is not in the state with lowest P (important for SSB.) The formulation of the theory in terms of the Green s functions is a new way of looking at light-front quantization at T = 0 and at T 0. We interpret GPDs as light-front density matrices. This is a natural quantum-mechanical generalization of the parton model. Especially for the wavefunction of a large nucleus, a statistical approach makes sense. We introduce the chemical potential in a Lorentz-invariant way. Thermodynamic quantities are accessible in DLCQ, which can be applied even at µ T. However, DLCQ yields much more information than needed.